So, when we find the Cartesian product of two spaces, the Cartesian product will be connected, right? Also, what will be the cartesian product if we just flipped it to $8 \times S^1$.(Here 8 is a number, same as joining two circles).
Asked
Active
Viewed 138 times
1
-
Here is a picture of $8\times S^1$. To comment your first question: the Cartesian product of two connected spaces will be connected. If one factor has several connectivity components so does the product. And, the mapping $(x,y)\mapsto (y,x)$ is always a homemorphism between $X\times Y$ and $Y\times X$. – Jyrki Lahtonen Sep 17 '17 at 18:18
-
This was really helpful. Does the opposite of that $S^1 \times 8$ the same? – contemplator Sep 17 '17 at 18:29